A005995 Alkane (or paraffin) numbers l(8,n). (Formerly M2916)

1, 3, 12, 28, 66, 126, 236, 396, 651, 1001, 1512, 2184, 3108, 4284, 5832, 7752, 10197, 13167, 16852, 21252, 26598, 32890, 40404, 49140, 59423, 71253, 85008, 100688, 118728, 139128, 162384, 188496, 218025, 250971, 287964, 329004, 374794, 425334, 481404, 543004

Offset

0,2

Comments

From M. F. Hasler, May 01 2009: (Start)

Also, number of 5-element subsets of {1,...,n+5} whose elements sum to an odd integer, i.e. column 5 of A159916.

A linear recurrent sequence with constant coefficients and characteristic polynomial x^9 - 3*x^8 + 8*x^6 - 6*x^5 - 6*x^4 + 8*x^3 - 3*x + 1. (End)

Equals (1/2)*((1, 6, 21, 56, 126, 252, ...) + (1, 0, 3, 0, 6, 0, 10, ...)), see A000389 and A000217.

Equals row sums of triangle A160770.

F(1,5,n) is the number of bracelets with 1 blue, 5 identical red and n identical black beads. If F(1,5,1) = 3 and F(1,5,2) = 12 taken as a base, F(1,5,n) = n(n+1)(n+2)(n+3)/24 + F(1,3,n) + F(1,5,n-2). [F(1,3,n) is the number of bracelets with 1 blue, 3 identical red and n identical black beads. If F(1,3,1) = 2 and F(1,3,2) = 6 taken as a base F(1,3,n) = n(n+1)/2 + [|n/2|] + 1 + F(1,3,n-2)], where [|x|]: if a is an integer and a<=x<a+1, [|x|]=a. - Ata A. Uslu and Hamdi G. Ozmenekse, Mar 16 2012

References

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Winston C. Yang (paper in preparation).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

N. J. A. Sloane, Classic Sequences

L. Smith, Polynomial invariants of finite groups. A survey of recent developments. Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 211-250.

Ata A. Uslu and Hamdi G. Ozmenekse, F(1,3,n)

Ata A. Uslu and Hamdi G. Ozmenekse, F(1,5,n)

Index entries for linear recurrences with constant coefficients, signature (3, 0, -8, 6, 6, -8, 0, 3, -1).

Formula

G.f.: (1+3*x^2)/((1-x)^3*(1-x^2)^3).

a(2n-1) = n(n+1)(n+2)(2n+1)(2n+3)/30, a(2n) = (n+1)(n+2)(n+3)(4n^2+6n+5)/ 30. [M. F. Hasler, May 01 2009]

a(n) = (1/240)*(n+1)*(n+2)*(n+3)*(n+4)*(n+5) + (1/16)*(n+2)*(n+4)*(1/2)*(1+(-1)^n). [Yosu Yurramendi, Jun 20 2013]

a(n) = A038163(n)+3*A038163(n-2). - R. J. Mathar, Mar 29 2018

Maple

a:= n-> (Matrix([[1, 0$6, -3, -9]]). Matrix(9, (i, j)-> if (i=j-1) then 1 elif j=1 then [3, 0, -8, 6, 6, -8, 0, 3, -1][i] else 0 fi)^n)[1, 1]: seq(a(n), n=0..40); # Alois P. Heinz, Jul 31 2008

Mathematica

a[n_?OddQ] := 1/240*(n+1)*(n+2)*(n+3)*(n+4)*(n+5); a[n_?EvenQ] := 1/240*(n+2)*(n+4)*(n+6)*(n^2+3*n+5); Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Mar 17 2014, after M. F. Hasler *)
LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {1, 3, 12, 28, 66, 126, 236, 396, 651}, 40] (* Ray Chandler, Sep 23 2015 *)

Prog

(PARI) a(k)= if(k%2, (k+1)*(k+3)*(k+5), (k+6)*(k^2+3*k+5))*(k+2)*(k+4)/240 \\ M. F. Hasler, May 01 2009

Crossrefs

Cf. A160770, A053132 (bisection), A271870 (bisection), A018210 (partial sums).

Sequence in context: A294418 A308669 A115549 * A351643 A034503 A026557

Adjacent sequences: A005992 A005993 A005994 * A005996 A005997 A005998

Keyword

nonn,easy

Author

N. J. A. Sloane, Winston C. Yang (yang(AT)math.wisc.edu)

Status

approved